On 𝐿_{𝑝}-Brunn-Minkowski type and 𝐿_{𝑝}-isoperimetric type inequalities for measures
Bibliographic record
Abstract
In 2011 Lutwak, Yang and Zhang extended the definition of the L p L_p -Minkowski convex combination ( p ≥ 1 p \geq 1 ) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in R n \mathbb {R}^n , and as a consequence, extended the L p L_p -Brunn-Minkowski inequality ( L p L_p -BMI) to the setting of all measurable sets. In this paper, we present a functional extension of their L p L_p -Minkowski convex combination—the L p , s L_{p,s} –supremal convolution and prove the L p L_p -Borell-Brascamp-Lieb type ( L p L_p -BBL) inequalities. Based on the L p L_p -BBL type inequalities for functions, we extend the L p L_p -BMI for measurable sets to the class of Borel measures on R n \mathbb {R}^n having ( 1 s ) \left (\frac {1}{s}\right ) -concave densities, with s ≥
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.008 | 0.021 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.004 |
| Bibliometrics | 0.003 | 0.005 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.007 | 0.017 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.003 | 0.013 |
| Insufficient payload (model declined to judge) | 0.018 | 0.005 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".